1989 General Insurance Convention The application of additive and multiplicative Generale Interactive Models (GLIM) in Health Insurance. Linear Paper for the XXI ASTIN Colloquium New York, 1989 subject: Empirical Investigations. Bob J.J. Alting von Geusau June 1989.
1 1. Introduction and Summary. For the private health insurance industry in the Netherlands a model is developped which can describe accurately the interaction between the various influencing factors which determine the size of health claims. Most of these factors such as sex of the insured, age of the insured, or type of coverage, can be considered as working multiplicatively on the base claim size, For other factors this seems less true: it might be that far them an additive model will be more appropriate. This paper provides a description of an example of a model which originally uses the multiplicative specifications but which is generalised easily to the use of additive factors. Only the basic technique is given, which means that some major points are still open: for instance when to use the purely multiplicative model and when the mixed one. The paper is concluded with some examples from real life.
2 2. The General Linear Interactive Model (GLIM): the multiplicative Gamma-case. If the claimcost per insured k are symbolized by the variable and the claimcost are supposed to be Gamma distributed, we can define as the density of : (1) The parameters µ. and. in (1) are also visible of course in (2) the expectation the variance In principle both sets of parameters can be of a very complicated structure, but in multiplicative GLIM we restrict ourself to the according to (3)
In formula (3) Xk is a vector with zeros and ones in such a way that the correct linear combination of elements from the rating vector is obtained. The second restriction affects : 3 (4) = for all k, where is the scale factor in the Gamma distribution: since at the moment we are only interested in the average value (3) we donot have to estimate the value of thin parameter. In order to estimate the values of the elements of sample z1,.., zk,..., zn and the log-likelihood function we use the (5) log = constant + since (3) and (4) hoids = constant + function(,zk) + k=1 As a function of (6) has its maximum when (6) for m=1,..,m, where xkm is element number m in the vector. The consequence of this definition is that for each claim we cannot have more than M rating factors.
4 Formula (6) is a system of M equations with H unknowns but since this system is not linear we need a little bit more than some basic knowledge of matrix algebra. Van Eeghen, Greup and Nijssen in "Ratemaking" (1983) describe the elegant method of scoring as a way to solve (6) as follows: Define the information matrix by: (7) This matrix is negative definite for the values (8) where is the nth approximation of the solution of (6). The method of scoring finally solves (6) by the iterative procedure (9) Formula (9) converges quite easily when the chosen initial value (10) is not too unreasonable.
It makes the situation easier that in this specific case (7) boils down to 5 (11) for each which means that it is necessary to compute the inverse matrix of (11) only once. 3. GLIM generalised to the mixed additive-multiplicative case. In health insurance it nay be necessary to generalise (3) to (12) (say) for the observations Zk with k=l,..,n where: the specific vector for observation k with zeros and ones for the additive factors, and the vector with the A additive factors. As an example of such an insured can be considered. additive factor the deductible per For the log-likelihood function this means: (13) log = constant + function
Therefore (6) becomes a system of (M+A) equations with (M+A) unknowns: 6 (14) for a=1,..,a and for m=1,..,m. which again cannot be solved directly. With scoring it takes some tedious calculations but finally we find for the information matrix the composition of the submatrices: for a=1,..,a and b=l,..,a and for m=l,..,m and p=l,..,m the matrices given by: of (15)* - Expectation (15)** - Expectation (15)*** - Expectation
7 which totalises into the information matrix by the recipe: (16) and leads finally to the scoring formula: (17) The convergence of (17) is much less convincing but in the cases I have analysed the solution came up after some time: the number of necessary iterations appeared to be about three times as high as in the purely multiplicative case. This is the price we have to pay for the fact that (16) is not strictly negative definite in all cases.
8 4. Some results on real data. The methods mentioned in the previous paragraphs have been applied to the 1987 and the 1988 data from private health insurance companies in the Netherlands. This data has been collected by the Foundation KISG from 3.5 million insureds with regards to the following activities: - the cost of hospitalisation - the cost of a specialist in a clinic - the cost of a specialist without hospitalisation - the cost of paramedical aid - the coat of other medical care, such as medical transportation, artificial legs and arms, dentures, obstetrical aid, home care, etcetera. Of these activities the gross and the net amounts per insured per year have been collected plus the information related to the the individual situation of the insured with regards to: - hospitalisation 3rd class or better - sex of the insured - area Where the insured is living: the highly populated Western part of the Netherlands is much more expensive than the North and the South - the age of the insured - the level of the deductible - the type of health insurance contract: in the Netherlands one can buy private health insurance through an individual contract or in some cases collectively via the employer in a group contract - the fact whether the cost of the family doctor is insured yes or no: it seems that this factor indicates a totally different type of risk. On this information I have applied the general GLIM-models in a score of variants of which I give three examples. The first one is a purely multiplicative one with compound age factors:
9 Base Claim per year net Dfls 667 664 Multiplicative factors in X: Year 1987 Year 1988 Hospitalisation Class: 3 l&2 114 112 Sex male female 99 Area: average cheap expensive 91 105 96 109 Age: 0-4 109 5-9 51 10-14 66 15-19 64 20-24 male idem female 25-29 male idem female 30-34 male idem female 35-39 male idem female 40-44 46-49 50-54 55-59 60-64 65-69 70-74 75+ 54 87 49 158 58 161 71 121 120 147 179 231 326 412 548 112 51 70 65 53 82 50 156 59 161 69 123 120 147 186 238 333 434 578 Deductible: low average high 95 58 88 57 Contract: individual collective 105 103 Family Doctor: included not included 84 83 Table 1. Estimation of the base claim per year and the multiplicative factors in X on the KISG 1987 and 1988 deck, with the introduction of separate multiplicative factors to sex for some ages.
10 An example of how table 1 works: let us take a gentleman of over 75: base claim multiplicative factors: * 2nd classe * male * expensive area * ago 75+ * low deductible * individually insured * family doctor included total nett claim per year fl. 667 fl. 4375 1,14 1 1,05 5,48 1 1 1 In a high deductible scenario the table let this amount shrink to 0,58 * 4375 = 2538, which is around two thousand guilders less. This difference seems to be too large. So in table 2 we use the mixed GLIM where the deductibles compounded with some age factors are tagged as additive factors. Deductible Age for male and female: 0-19 20-39 40-59 60+ low: under fl. 250 per year 1987: 180 173 205 836 1988: 189 22 249 729 average: fl. 250-750 per year 1987: 178 81 178 1068 1998: 150 73-155 547 high: over fl. 750 per year 1987: 0 22-116- 324 1988: 0 178-91- 10 Table 2a. Additive constants in the mixed GLIM-model for the claimfactor 'Deductible' on KISG 1987 and 1988: these constants can only be used in connection with table 2b.
11 Year 1987 Base claim per year Dfls. 477 Year 1988 445 Multiplicative factors in X: Hospitalisation Class: 3 1&2 118 115 Sex : male female 101 101 Area: average cheap 89 95 expensive 106 108 Age: 0-4 5-9 10-14 16-19 20-24 male idem female 25-29 male idea female 30-34 male idem female 35-39 male idem female 40-44 45-49 50-64 55-59 60-64 65-69 70-74 75+ 102 37 51 50 118 39 169 50 167 64 120 121 154 195 131 254 369 560 112 39 59 54 83 118 77 211 88 214 164 121 157 208 195 323 464 674 Contract: individual collective 106 Family Doctor: included not included 79 104 80 Table 2b. Estimation of the base claim in guilders per year and the multiplicative factors in X on KISG 1987 and 1988: this table can only be used in connection with table 2a.
Back to our example to demonstrate how both tables are used together: base claim 1987 fl. 477 12 multiplicative factors: * 2e class * male * expensive area * age 75+ * individual cover * family doctor included sub result additive factor: + low deductible, age 75+ total net claim in 1987 fl. 3341 836 fl. 4177 1,18 1 1,06 5,60 1 1 With a high deductible the amount would have been fl. 3665 - that is 512 guilders lower - and that seems more plausible. This better fit is also indicated by the Power mean squared error: in table 2a/2b this reads 685, while in table 1 it is 734. Finally I have listed table 3, where the multiplicative GLIM is demonstrated on a separate insurance activity: the cost of hospitalisation in 1987. The introduction of separate age/sex factors leads to interesting results:
13 Base claim in 1987 283 Multiplicative factors in X: Hospitalisation class: Sex : Area: 3 l & 2 1 1 3 male female 90 average cheap 91 expensive 103 male female Age: 0-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 46-49 50-54 55-59 60-64 65-69 70-74 759 61 66 81 131 164 253 358 570 739 7 189 60 56 82 91 184 191 153 164 196 221 257 347 488 668 1024 Deductible: Contract: Family doctor: Standaard deviation MSE: low average 98 high 65 individual group 102 included not included 82 605 Table 3. Estimation of the base claim per year in guilders and the multiplicative factors in X for the cost of hospitalisation in 1987 with the introduction of separate multiplicative factors to sex for some ages.
14 5. Resume. Pour la description d'un portefeuille Couts Medicaux la methode GLIM (General Linear Interactive Model) peut donner des instruments utilisables. I1 parait possible de generaliser cette méthode avec des variantes mixtes du genre additif-multiplicatif. Finallement le papier donne quelques exemples. Bob J.J. Alting von Geusau, Juin 1989 - June 1989,